For instance we’ve got a $ok$ group scheme $G$ performing on a $ok$ scheme $X$, we are able to take into account its quotient within the class of stacks, the same old definition of it could be the quotient stack $[X/G]$ outlined by
$[X/G](S)$ is the groupoid of $G$ bundles over $S$ collectively an equivariant maps to $X$.
My Query 1 is why do not we take the specific quotient?
I feel the specific quotient exists as a result of we are able to take the specific quotient within the classes of 2-functors from Schemes to Groupoids, after which sheafify (stackify).
I perceive that the stack quotients get pleasure from some superb properties, e.g. it makes each motion appear to be a free motion and it the idea of sheaves downstair is equal to the idea of sheaves.
however why individuals (or do they) do not research categorical quotient as effectively?
My Query 2 is how do stack quotients differentiate from categorical quotients?
Stack quotient is the same as the specific quotient within the case free motion, and I’ve the sensation that the 2 notions agree solely when the motion is free, is it true?
Thanks very a lot!